A116950 Number of functional patterns on n elements; or digraphs with maximum outdegree 1, n arrows and every point connected to an arrow.

1, 2, 7, 20, 61, 174, 514, 1478, 4303, 12437, 36084, 104494, 303167, 879283, 2552803, 7413583, 21544347, 62635823, 182199853, 530228946, 1543761513, 4496523995, 13102414665, 38193626823, 111375529695, 324891970936, 948051861938, 2767336312386, 8080206646244

Offset: 0

Author

Franklin T. Adams-Watters, Mar 29 2006

Keywords

• easy
• nice
• nonn

Comments

A001372 counts functional patterns from a set with n elements to itself; A000041 (partition function) counts functional patterns from a set with n elements to a disjoint set; this is the general case where the range may overlap the domain but may also include other values.

Examples

			For n=2 there are the following 7 digraphs:
o-+.o-+ o->o-+ o->o o-+.o->o o->o->o o->o o->o
^.|.^.| ...^.| ^..| ^.|..... ....... ...^ ....
+-+.+-+ ...+-+ +--+ +-+..... ....... o--+ o->o
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2127

Crossrefs

Cf. A000041, A001372, A002861, A000081.

Programs

Mathematica

nmax = 750;
A002861 = Cases[Import["https://oeis.org/A002861/b002861.txt", "Table"], {, }][[;; nmax + 2, 2]];
A000081 = Cases[Import["https://oeis.org/A000081/b000081.txt", "Table"], {, }][[;; nmax + 2, 2]];
etr[p_] := Module[{b}, b[n_] := b[n] = If[n == 0, 1, Sum[Sum[d*p[d], {d, Divisors[j]}]*b[n - j], {j, 1, n}]/n]; b];
b[n_] := A002861[[n]] + A000081[[n + 2]];
a = etr[b];
a[0] = 1;
a /@ Range[0, nmax](* Jean-François Alcover, Mar 13 2020 *)

Formula

Euler transform of A002861(n) + A000081(n+1).

a(n) ~ c * d^n / sqrt(n), where d = A051491 = 2.95576528565199497471481752412..., c = 3.435908969217935496995961718... . - Vaclav Kotesovec, Sep 10 2014